A generalization of the zero-divisor graph for modules

Katayoun Nozari, Shiroyeh Payrovi

Let $R$ be a commutative ring and $M$ a Noetherian $R$-module. The zero-divisor graph of $M$, denoted by $\Gamma(M)$, is an undirected simple graph whose vertices are the elements of $Z_R(M)\setminus\operatorname{Ann}_R(M)$ and two distinct vertices $a$ and $b$ are adjacent if and only if $abM= 0$. In this paper, we study diameter and girth of $\Gamma(M)$. We show that the zero-divisor graph of $M$ has a universal vertex in $Z_R(M)\setminus r(\operatorname{Ann}_R(M))$ if and only if $R=\ZZ_2\oplus R'$ and $M=\ZZ_2\oplus M'$, where $M'$ is an $R'$-module. Moreover, we show that if $\Gamma(M)$ is a complete graph, then one of the following statements is true: \begin{itemize} ıem[(i)] $\operatorname{Ass}_R(M)=\{\frak m_1,\frak m_2\}$, where $\frak m_1,\frak m_2$ are maximal ideals of $R$. ıem[(ii)] $\operatorname{Ass}_R(M)=\{\frak p\}$, where $\frak p^2\subseteq\operatorname{Ann}_R(M)$. ıem[(iii)] $\operatorname{Ass}_R(M)=\{\frak p\}$, where $\frak p^3\subseteq\operatorname{Ann}_R(M)$. \end{itemize}